1. Determine how each function is related to graph of f(x) = |x|.  f(x) = 2|x|  f(x) = |x-1|  f(x) = |x| + 3

3. Greatest Integer Function – the most notable step function Step Functions – functions whose graphs resemble sets of stair steps. Notation of Greatest Integer Function : x Meaning of Greatest Integer Function : the greatest integer less than or equal to x.

4. 012345678-2-3-4-5-6-7 It may be helpful to visualize this function a little more clearly by using a number line. When you use this function, the answer is the integer on the immediate left on the number line. 6.31-6.31 Exception: When you evaluate an exact integer, like 3, the answer is the integer itself.

5. Let’s evaluate some greatest integers… 2 = 9 = -3 = 2 ? 9 ? ? the greatest integer less than or equal to x.

6. Let’s evaluate some greatest integers… 2.2 = 1/21/2 = -4.1 = 2? 0 ? -5 ? the greatest integer less than or equal to x. 0 1 2 3 -2 -3 -4 -5

7. Let’s evaluate some greatest integers… 9.1 = 51/351/3 = -2 2 / 9 = 9 ? 5 ? -3 ? the greatest integer less than or equal to x. 0 1 2 3 -2 -3 -4 -5

8. If there is an operation inside the greatest integer brackets, it must be performed before applying the function.

9. xy 0.25.5.75 1 1.25 1.5 1.75 2 Now that you know how to evaluate greatest integer functions, you can graph them. y = x +2

10. xy 0.25.5.75 1 1.25 1.5 1.75 2 Now that you know how to evaluate greatest integer functions, you can graph them. y = x - 4

11. xy 0.25.5.75 1 1.25 1.5 1.75 2 Now that you know how to evaluate greatest integer functions, you can graph them. y = -x + 2

12. When all these points are strung together the graph looks something like a series of steps. Reasoning for: ‘STEP FUNCTION’. Notice that the left of each step begins with a closed point but the right of each step ends with an open point We can’t really state the last x- value on each step because there is always another to the right of the last one you may name. So instead we describe the first x- value that is NOT on a given step. Example: (1,0)

13. Rather than place a long series of points on the graph, a line segment can be drawn for each step as shown to the right. The graphs shown thus far have been magnified to make a point. However, these graphs are usually shown at a normal scale. f(x) = [x]

14. Wheels Bike Rentals charges a $6.00 flat rate and $1.50 for each hour you rent a bicycle including fractions of an hour (For example, 3.5 hours is $1.50(3) + $6.00). Use the greatest integer function to create a model for the cost C of renting a bicycle for x hours. Sketch the graph for up to 5 hours. xC